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(continued from front flap) mAthemAtiCS $26.99 US / $30.00 CAn S t e ≥ Δ Φ ∞ Φ Δ Φ ∞ Φ Δ ≤ P r a i s e f or Ian Stewart wa n P u r s u i t of the Unknow ≥ Δ Φ ∞ Φ Δ Φ ∞ Φ Δ ≤ r I n M An approachable, lively, and informative guide to the t ost people are familiar with history’s mathematical building blocks that form the foundations of great equations: Pythagoras’s theorem, modern life, In Pursuit of the Unknown is also a penetrating 1 7 E q uations That for instance, or Newton’s Law of Gravity, “Stewart has a genius for explanation. . . . Mathematics doesn’t come more entertaining than this.” exploration of how we have long used equations to make —New ScieNtiSt C h a n ged the World or Einstein’s theory of relativity. But the way these sense of, and in turn influence, our world. mathematical breakthroughs have contributed to human t I hn progress is seldom appreciated. In his new work, In e “Combines a deep understanding of math with an engaging literary style.” P Pursuit of the Unknown, celebrated mathematician Ian U —the waShiNgtoN PoSt u Stewart untangles the roots of our most important n r ks mathematical statements to show that equations have long nu “Possibly mathematics’ most energetic evangelist.” oi been a driving force behind nearly every aspect of our lives. t w o Using seventeen of our most crucial equations, —the SPectator (London) nf Stewart illustrates that many of the advances we now take for granted—in science, philosophy, technology, and “Stewart is able to write about mathematics for general readers. He can make tricky ideas art beyond—were made possible by mathematical discoveries. w simple, and he can explain the maths of it with aplomb. . . . Stewart admirably captures e St For example, the Wave Equation allowed engineers to vril compelling and accessible mathematical ideas along with the pleasure of A measure a building’s response to earthquakes, saving © thinking about them. He writes with clarity and precision.” C 1 6.25” x 9.5” h 7 countless lives; without the Wave Equation, moreover, ian Stewart —LoS aNgeLeS timeS a E S: 1-1/8” is Emeritus Professor of n q scientists would never have discovered electromagnetic B: 7/8” g u e waves, which in turn led to the invention of radio and a Mathematics, active researcher at Warwick University in d BASIC t England, and author of many books on mathematics. His “A highly gifted communicator, able not only to explain the motivation of mathematicians th io television. The equation at the heart of information HC n down the centuries but to elucidate the resulting mathematics with both clarity e theory, devised by Claude Shannon, forms the basis for s writing has also appeared in publications including New W 4/COLOR T and style. The whole is leavened by his inimitable understated wit.” modern digital communication systems, which have Scientist, Discover, and Scientific American. He lives in o h r a FINISH: —the timeS educatioN SuPPLemeNt l revolutionized everything from politics to business to Coventry, England. d t MATTE POLY interpersonal relationships. And the Black-Scholes model, SPOT GLOSS ON EQUATIONS ON $26.99 US / $30.00 CAN used by bankers to track the prices of financial derivatives FRONT COVER ISBN 978-0-465-02973-0 over time, led to massive growth in the financial sector, 52699 Ian Stewa r t Jacket design by Jennifer Carrow A Member of the Perseus Books Group thereby contributing to the banking crisis of 2008—the 03/12 www.basicbooks.com effects of which we are still feeling today. 9 780465 029730 (continued on back flap) IN PURSUIT OF THE UNKNOWN Also by Ian Stewart: Concepts of Modern Mathematics Game, Set, and Math The Problems of Mathematics Does God Play Dice? Another Fine Math You’ve Got Me into Fearful Symmetry(with Martin Golubitsky) Nature’s Numbers From Here to Infinity The Magical Maze Life’s Other Secret Flatterland What Shape Is a Snowflake? The Annotated Flatland Math Hysteria The Mayor of Uglyville’s Dilemma Letters to a Young Mathematician Why Beauty Is Truth How to Cut a Cake Taming the Infinite/The Story of Mathematics Professor Stewart’s Cabinet of Mathematical Curiosities Professor Stewart’s Hoard of Mathematical Treasures Cows in the Maze Mathematics of Life with Terry Pratchett and Jack Cohen The Science of Discworld The Science of Discworld II: the Globe The Science of Discworld III: Darwin’s Watch with Jack Cohen The Collapse of Chaos Figments of Reality Evolving the Alien/What Does a Martian Look Like? Wheelers(science fiction) Heaven(science fiction) IN PURSUIT OF THE UNKNOWN 17 Equations That Changed the World ✺ IAN STEWART A Member of the Perseus Books Group New York Copyright © 2012 by Ian Stewart Published in the United States in 2012 by Basic Books, A Member of the Perseus Books Group Published in Great Britain in 2012 by Profile Books All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 387 Park Avenue South, New York, NY 10016-8810. Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail [email protected]. A CIP catalog record for this book is available from the Library of Congress. LCCN: 2011944850 ISBN: 978-0-465-02973-0 10 9 8 7 6 5 4 3 2 1 Contents WhyEquations?/viii 1 Thesquawonthehippopotamus/1 Pythagoras’sTheorem 2 Shorteningtheproceedings/21 Logarithms 3 Ghostsofdepartedquantities/35 Calculus 4 Thesystemoftheworld /53 Newton’sLawofGravity 5 Portentoftheidealworld/73 TheSquareRootofMinusOne 6 Muchadoaboutknotting/89 Euler’sFormulaforPolyhedra 7 Patternsofchance/107 NormalDistribution 8 Goodvibrations/131 WaveEquation 9 Ripplesandblips/149 FourierTransform 10 Theascentofhumanity/165 Navier–StokesEquation 11 Wavesintheether/179 Maxwell’sEquations 12 Lawanddisorder/195 SecondLawofThermodynamics ProfileBooks-SeventeenEquations DataStandardsLtd,Frome,Somerset–5/12/2011 00Seventeen_Prelims.3d Page5of10 vi Contents 13 Onethingisabsolute/217 Relativity 14 Quantumweirdness /245 Schro¨dinger’sEquation 15 Codes,communications,andcomputers/265 InformationTheory 16 Theimbalanceofnature/283 ChaosTheory 17 TheMidasformula/295 Black–ScholesEquation WhereNext?/317 Notes/321 IllustrationCredits/330 Index/331 ProfileBooks-SeventeenEquations DataStandardsLtd,Frome,Somerset–5/12/2011 00Seventeen_Prelims.3d Page6of10 To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe:=======, bicause noe .2. thynges, can be moare equalle. Robert Recorde, The Whetstone of Witte, 1557 ProfileBooks-SeventeenEquations DataStandardsLtd,Frome,Somerset–5/12/2011 00Seventeen_Prelims.3d Page7of10 Why Equations? E quations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. However, equations have a reputation for being scary: Stephen Hawking’s publishers told him that every equation would halve the sales of A Brief History of Time, but then they ignored their own advice and allowed him to include E = mc2 when cutting it out would allegedly have sold another 10 million copies. I’m on Hawking’s side. Equations are too importanttobehiddenaway.Buthispublishershadapointtoo:equations are formal and austere, they look complicated, and even those of us who love equations can be put off if we are bombarded with them. Inthisbook,Ihaveanexcuse.Sinceit’saboutequations,Icannomore avoid including them than I could write a book about mountaineering withoutusingtheword‘mountain’.Iwanttoconvinceyouthatequations have played a vital part in creating today’s world, from mapmaking to satnav, from music to television, from discovering America to exploring themoonsofJupiter.Fortunately,youdon’tneedtobearocketscientistto appreciate the poetry and beauty of a good, significant equation. Therearetwokindsofequationsinmathematics,whichonthesurface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it – to make the unknown known. The distinction is not clear-cut, because sometimes the same equation can be usedinbothways,butit’sausefulguideline.Youwillfindbothkindshere. Equations in pure mathematics are generally of the first kind: they revealdeepandbeautifulpatternsandregularities.Theyarevalidbecause, given our basic assumptions about the logical structure of mathematics, there is no alternative. Pythagoras’s theorem, which is an equation expressed in the language of geometry, is an example. If you accept Euclid’s basic assumptions about geometry, then Pythagoras’s theorem is true. Equations in applied mathematics and mathematical physics are usually of the second kind. They encode information about the real ProfileBooks-SeventeenEquations DataStandardsLtd,Frome,Somerset–5/12/2011 00Seventeen_Prelims.3d Page8of10 WhyEquations? ix world;theyexpresspropertiesoftheuniversethatcouldinprinciplehave been very different. Newton’s law of gravity is a good example. It tells us howtheattractiveforcebetweentwobodiesdependsontheirmasses,and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe. But Newton’slawisn’tamathematicaltheorem;it’strueforphysicalreasons,it fitsobservations.Thelawofgravitymighthavebeendifferent.Indeed,itis different: Einstein’s general theory of relativity improves on Newton by fitting some observations better, while not messing up those where we already know Newton’s law does a good job. Thecourseofhumanhistoryhasbeenredirected,timeandtimeagain, byanequation.Equationshavehiddenpowers.Theyrevealtheinnermost secrets of nature. This is not the traditional way for historians to organise the rise and fall of civilisations. Kings and queens and wars and natural disasters abound in the history books, but equations are thin on the ground. This is unfair. In Victorian times, Michael Faraday was demonstrating connections between magnetism and electricity to audiences at the Royal Institution in London. Allegedly, Prime Minister William Gladstone asked whether anything of practical consequence wouldcomefromit.Itissaid(onthebasisofverylittleactualevidence,but whyruinanicestory?)thatFaradayreplied:‘Yes,sir.Onedayyouwilltax it.’Ifhedidsaythat,hewasright.JamesClerkMaxwelltransformedearly experimental observations and empirical laws about magnetism and electricity into a system of equations for electromagnetism. Among the many consequences were radio, radar, and television. Anequationderivesitspowerfromasimplesource.Ittellsusthattwo calculations, which appear different, have the same answer. The key symbolistheequalssign,¼.Theoriginsofmostmathematicalsymbolsare eitherlostinthemistsofantiquity,oraresorecentthatthereisnodoubt where they came from. The equals sign is unusual because it dates back morethan450years,yetwenotonlyknowwhoinventedit,weevenknow why.TheinventorwasRobertRecorde,in1557,inTheWhetstoneofWitte. He used two parallel lines (he used an obsolete word gemowe, meaning ‘twin’)toavoidtediousrepetitionofthewords‘isequalto’.Hechosethat symbolbecause‘notwothingscanbemoreequal’.Recordechosewell.His symbol has remained in use for 450 years. The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds,andanexternalphysicalreality.Equationsmodeldeeppatternsin the outside world. By learning to value equations, and to read the stories ProfileBooks-SeventeenEquations DataStandardsLtd,Frome,Somerset–5/12/2011 00Seventeen_Prelims.3d Page9of10